sin x 2 Imaginary Trigonometry
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sin x = 2: IMAGINARY TRIGONOMETRY. 191. culminating in the theory of functions with its numerous appli-. cations in nearly every branch of mathematics, …
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sinx = 2: Imaginary Trigonometry
De Moivre’s theorem was the key to a whole new world
of imaginary or complex trigonometry.
-Herbert Mc Kay, The World of Numbers
(1946), p. 157
Imagine you have just bought a brand new hand-held calculator
and find to your dismay that when you try to subtract 5 from 4,
you get an error sign. Yet this is precisely the situation in which
first-grade pupils would find themselves if their teacher asked
them to take away five apples from four: “It cannot be done!”
would be the class’s predictable response.
The history of mathematics is full of attempts to break the
barrier of the “impossible.” Many of these attempts have ended
in failure: for more than two thousand years mathematicians
tried to find a construction, using straightedge and compass
alone, that would trisect an arbitrary angle-until it was proved,
around the middle of the nineteenth century, that such a construction
is impossible. The countless attempts to “square the
circle”-to construct, again with straightedge and compass
alone, a square whose area equals that of a given circle-have
likewise proved futile (which does not prevent…
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